Question

Prove that time period of a satellite is given by T = root of 3pi / G × rho.

Answer 1

Answer:

Given:

Satellite travels aound a planet.

To Prove:

Time period is equal to :

Concept:

∴ Time = total distance/speed

Let radius be r , density = ρ , universal gravitational constant be G.

In the case, the total distance is the circumference = 2πr , and the speed is the orbital velocity of satellites = √(Gm/r)

Calculation:

$\therefore \: time$

$= \dfrac{distance}{speed}$

$= \dfrac{2\pi r}{ \sqrt{ \frac{Gm}{r} } }$

Now splitting mass = density × volume and considering the earth to be perfe the sphere :

$= \dfrac{2\pi r}{ \sqrt{ \dfrac{G \times (\frac{4}{3}\pi {r}^{3} \times \rho) }{r} } }$

After cancelling all the necessary terms:

$= \dfrac{3\pi}{ \sqrt{(G \times \rho)} }$

So proved .

Answer 2

Answer:T=√3π/Grhi

Explanation: